The striker pounced in the 50th minute as Colin Calderwood's side took advantage of the fact that promotion-rivals Doncaster's game against Leeds was called off.
It could have been a bigger win for Forest as they carved out a host of second half-chances with Junior Agogo guilty of a spectacular miss when presented with an open goal.
But Forest rarely looked under threat from shot-shy Swindon as they maintained their push for a return to the automatic promotion places.
Swindon keeper Petr Brezovan almost handed Forest an early advantage as he hesitated in possession after collecting a back pass from Jerel Ifil but, while Tyson was quick to close him down, the ball deflected fortuitously off the striker and straight back into the grateful arms of the keeper.
At the other end, Forest keeper Paul Smith had to keep a careful eye on a long-range effort from Anthony McNamee as it flew wide, before a clever lob from Jon-Paul McGovern dropped the wrong side of the post.
Tyson's pace was causing Swindon problems at the back and when he ran at Ifil, the defender was forced into a lunging challenge right on the edge of the box which earned him a booking.
Forest actually survived almost ten first-half minutes with ten men as Wes Morgan went off the pitch to have stitches inserted in a head wound.
Agogo, who has just returned from the Africa Cup of Nations with Ghana, launched himself into an explosive run that was only brought to a halt by an obstruction from Ifil on the edge of the box.
And from the free-kick, Lewis McGugan saw his initial effort bounce back off the wall but forced a spectacular save from Brezovan as he lashed the rebound goalwards with incredible power.
The first half had seen Forest dominate but fail to deliver the killer pass when it counted.
But five minutes after the break they finally found their accuracy as Matt Lockwood fed Tyson down the left-hand channel.
The striker's pace took him beyond the last man and in confident fashion he fired low into the bottom corner.
And it was the first of a flurry of chances for the home side, who saw Tyson flash a second effort narrowly wide, Kris Commons force a fine save from Brezovan with a vicious drive and McGugan fired over after his quick feet created the room for a shot.
Agogo missed a golden opportunity following neat build-up between Commons and Tyson with the Ghana frontman failing to properly connect when presented with an open goal.
A Craig Easton header tested Smith's alertness in the Forest goal, but it was still the home side on top with Chris Cohen and Agogo failing to hit the target with further chances.